Problem 91 - Physics

Question

Mercury is poured into a tall glass. Ethyl alcohol is then poured on top of the mercury until the height of the ethyl alcohol itself is \(110 \mathrm{~cm}\). The two fluids do not mix, and the air pressure at the top of the ethyl alcohol is one atmosphere. What is the absolute pressure at a point that is \(7.10 \mathrm{~cm}\) below the ethyl alcohol-mercury interface?

Step-by-Step Solution

Answer

The absolute pressure at the point is approximately 119.3 kPa.

Step 1: Understand the Problem

We need to calculate the absolute pressure at a point 7.10 cm below the interface of ethyl alcohol and mercury in a container with air pressure at the top of the ethyl alcohol being one atmosphere (1 atm).

Step 2: Determine Atmospheric Pressure

The atmospheric pressure at the top of the ethyl alcohol is 1 atm, which is equivalent to \(101,325\) Pascals (Pa). This will contribute to the total pressure at the point of interest.

Step 3: Calculate Pressure Due to Ethyl Alcohol

Ethyl alcohol has a density of approximately \(789\, \text{kg/m}^3\). The height of ethyl alcohol is 110 cm, which is 1.10 m in SI units. Use the hydrostatic pressure formula to calculate the pressure due to the column of ethyl alcohol: \[P_{ ext{ethyl}} = \rho_{ ext{ethyl}} \cdot g \cdot h_{ ext{ethyl}} = 789 \cdot 9.81 \cdot 1.10\] This simplifies to \(P_{ ext{ethyl}} \approx 8,521.33\, \text{Pa}\).

Step 4: Calculate Pressure Due to Mercury

Now, consider the pressure exerted by the mercury column below the ethyl alcohol. Mercury has a density of approximately \(13,600\, \text{kg/m}^3\). The depth of interest (7.10 cm below the interface) is converted to meters, which is 0.071 m. Calculate the pressure using the formula:\[P_{ ext{mercury}} = \rho_{ ext{mercury}} \cdot g \cdot h_{ ext{mercury}} = 13,600 \cdot 9.81 \cdot 0.071\]This simplifies to \(P_{ ext{mercury}} \approx 9,454.56\, \text{Pa}\).

Step 5: Calculate Total Absolute Pressure

The absolute pressure at 7.10 cm below the interface is the sum of the atmospheric pressure, the pressure exerted by the ethyl alcohol, and the pressure exerted by the mercury:\[P_{ ext{total}} = P_{ ext{atm}} + P_{ ext{ethyl}} + P_{ ext{mercury}} = 101,325 + 8,521.33 + 9,454.56\]This yields \(P_{ ext{total}} \approx 119,300.89\, \text{Pa}\) or approximately 119.3 kPa.

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases with the depth of the fluid.
When calculating hydrostatic pressure, you use the formula:

• \( P = \rho \cdot g \cdot h \)
• \( \rho \) is the fluid's density (mass per unit volume)
• \( g \) is the acceleration due to gravity (approximately \(9.81 \, \mathrm{m/s^2}\) on Earth)
• \( h \) is the height or depth of the fluid column above the point you are measuring

The hydrostatic pressure increases as you go deeper into a fluid.
In our exercise, both the ethyl alcohol and the mercury exert hydrostatic pressure on the point 7.10 cm below their interface.

Density

Density is a measure of mass per unit volume and it plays a critical role in determining hydrostatic pressure.
It is denoted by the Greek letter \( \rho \) and is usually expressed in kilograms per cubic meter (kg/m³).
Different substances have different densities, which influence the pressure exerted by fluid columns.For instance, in our exercise, ethyl alcohol has a density of approximately \(789 \, \text{kg/m}^3\), while mercury has a much higher density of approximately \(13,600 \, \text{kg/m}^3\).
This means that for the same depth, mercury will exert more pressure than ethyl alcohol due to its greater density.

Absolute Pressure

Absolute pressure is the total pressure exerted by a fluid, including atmospheric pressure and hydrostatic pressure.
It is an important concept because it tells us the true pressure at a point within a fluid. To find absolute pressure, you sum up:

• Atmospheric pressure, which is the pressure exerted by the weight of the atmosphere above the fluid
• Hydrostatic pressure of any fluid column above the point of interest

In our example, to find the absolute pressure 7.10 cm below the interface, we consider the pressures from both the ethyl alcohol and mercury columns, as well as the atmospheric pressure above the ethyl alcohol.

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the Earth's atmosphere at any given point.
It is a crucial part of understanding total pressure in fluid systems since every place on Earth below sea level experiences this pressure in a fluid system. At sea level, atmospheric pressure is approximately 101,325 Pascals or 1 atmosphere (atm).
This pressure varies with altitude and weather, but for most calculations, it's assumed to be constant.
In our exercise, this atmospheric pressure acts on top of the ethyl alcohol column, contributing to the total pressure at the point of interest below the fluid interface.